Faraday's Law Calculator
Calculate the induced EMF (voltage) from a changing magnetic flux using Faraday's law. Enter the number of coil turns, the flux change ΔΦ, and the time interval Δt.
Find the Induced EMF from Changing Magnetic Flux
Faraday's Law Formula
Where:
- \(\varepsilon\) — induced electromotive force, EMF (volts, V)
- \(N\) — number of turns of the coil
- \(\Phi\) — magnetic flux through one turn of the coil (webers, Wb)
- \(\Delta\Phi\) — change in magnetic flux, \(\Delta\Phi = |\Phi_2 - \Phi_1|\) (Wb)
- \(\Delta t\) — time interval over which the flux changes (seconds, s)
The formula states that the EMF induced in a coil is proportional to how fast the magnetic flux through it changes: more turns and a faster flux change produce a larger voltage. The minus sign is Lenz's law — the induced EMF always opposes the change in flux that created it.
Faraday's Law – Calculation Example
Example: a 100-turn coil sits in a magnetic field. The flux through the coil drops by 5 mWb over a period of 0.1 s (for example, when a magnet is pulled away from the coil). What EMF is induced?
\[ \Delta\Phi = 5\ \text{mWb} = 0.005\ \text{Wb}, \qquad \Delta t = 0.1\ \text{s} \]
\[ |\varepsilon| = N \frac{\Delta\Phi}{\Delta t} = 100 \times \frac{0.005}{0.1} \]
Result: \( |\varepsilon| = 5\ \text{V} \) (5,000 mV). The coil develops an average induced EMF of 5 volts while the flux is changing.
How to Use the Faraday's Law Calculator
- Enter the Number of Turns (N) of your coil. This is usually a whole number, e.g. 100.
- Enter the Change in Magnetic Flux (ΔΦ) — the absolute difference between the final and initial flux, \( |\Phi_2 - \Phi_1| \). Choose the unit: µWb, mWb, or Wb.
- Enter the Time Interval (Δt) over which the change occurs, and select its unit: ms, s, or min.
- Read the Induced EMF in volts (V) and In Millivolts (mV), together with the full calculation details.
What Is Faraday's Law of Induction?
Faraday's law of induction, discovered by Michael Faraday in 1831, is one of the cornerstones of electromagnetism. It states that a changing magnetic flux through a circuit induces an electromotive force — a voltage — in that circuit. No voltage appears if the flux is constant: something must change. The change can come from moving a magnet toward or away from a coil, moving the coil in a magnetic field, varying the field strength over time, or rotating the coil (which is exactly how generators work).
Magnetic flux \( \Phi \) measures how much magnetic field passes through a surface. It equals \( \Phi = B A \cos\theta \), where \( B \) is the magnetic flux density (in tesla), \( A \) is the area of the coil, and \( \theta \) is the angle between the field and the normal to the surface. Its unit, the weber, satisfies \( 1\ \text{Wb} = 1\ \text{T}\cdot\text{m}^2 = 1\ \text{V}\cdot\text{s} \).
The result \( |\varepsilon| = N\Delta\Phi/\Delta t \) is the average EMF over the time interval. If the flux changes at a constant rate, the average and the instantaneous EMF are the same. If the change is non-uniform (for instance, a sinusoidal flux in an AC generator), the instantaneous EMF at each moment is \( N|d\Phi/dt| \), which can be larger or smaller than the average.
The minus sign in the full law encodes Lenz's law: the induced current flows in the direction that opposes the change in flux. This is why the EMF is negative when the flux increases and positive when it decreases. This calculator returns the magnitude of the EMF, which is usually what you need for solving problems — the direction follows from Lenz's law once you know whether the flux is growing or shrinking.
Assumptions, Limitations and Practical Tips
- Average EMF only: the calculator uses the finite-change form \( N\Delta\Phi/\Delta t \). For rapidly or non-uniformly varying flux, treat the result as the average over the interval, not the instantaneous value.
- Uniform flux through the coil: every turn is assumed to link the same flux \( \Phi \). In tightly wound, compact coils this is a very good approximation.
- Self-inductance is ignored: the calculator computes the EMF from the external flux change. In real coils, the induced current creates its own flux (self-inductance), which slightly alters the observed voltage.
- Flux change is what matters: if \( \Delta\Phi = 0 \), no EMF is induced — a stationary coil in a constant field produces nothing, no matter how strong the field is.
- Check your units: mixing units is the most common source of errors. 1 mWb = 10−3 Wb and 1 µWb = 10−6 Wb; 1 ms = 10−3 s. The calculator converts everything to webers and seconds automatically.
Faraday's law is the working principle behind electric generators, transformers, induction cooktops, wireless charging, and metal detectors. If you are exploring other foundational physics laws, try our Hooke's Law calculator for elastic forces, the Point Charge Electric Field calculator for electrostatics, or the Gravitational Force calculator for Newtonian attraction.
Quick Reference: Typical Values
| Scenario | N | ΔΦ | Δt | Induced EMF |
|---|---|---|---|---|
| Magnet pulled from a small coil | 100 | 1 mWb | 0.05 s | 2 V |
| Generator coil rotating once per cycle | 500 | 10 mWb | 0.02 s | 250 V |
| Slowly changing transformer core flux | 1,000 | 2 mWb | 0.1 s | 20 V |
| Fast flux pulse in a sensor coil | 50 | 0.2 mWb | 5 ms | 2 V |
Frequently Asked Questions
How do I calculate induced EMF by hand?
Multiply the number of coil turns by the flux change, then divide by the time it took: \( |\varepsilon| = N \cdot \Delta\Phi / \Delta t \). First convert the flux to webers and the time to seconds. For example, with \( N = 200 \), \( \Delta\Phi = 3\ \text{mWb} = 0.003\ \text{Wb} \) and \( \Delta t = 0.2\ \text{s} \), you get \( |\varepsilon| = 200 \times 0.003 / 0.2 = 3\ \text{V} \).
What if I only know the magnetic field strength, not the flux?
Compute the flux first using \( \Phi = B A \cos\theta \), where \( B \) is the flux density in tesla, \( A \) is the coil area in square meters, and \( \theta \) is the angle between the field and the coil's normal. Then take the difference between the initial and final flux as \( \Delta\Phi \). For a circular coil of radius \( r \), the area is \( A = \pi r^2 \).
Why does a stationary coil in a strong magnetic field produce no voltage?
Because Faraday's law depends on the rate of change of flux, not its magnitude. A coil sitting in a constant field of any strength has \( \Delta\Phi = 0 \), so the induced EMF is exactly zero. Voltage only appears when something changes: the magnet moves, the field strength varies, the coil rotates, or the coil enters or leaves the field region.
How do I convert between Wb, mWb and µWb (and tesla)?
Use \( 1\ \text{Wb} = 1000\ \text{mWb} = 1{,}000{,}000\ \mu\text{Wb} \), so 1 mWb = 0.001 Wb and 1 µWb = 0.000001 Wb. The weber also equals \( 1\ \text{T}\cdot\text{m}^2 \) and \( 1\ \text{V}\cdot\text{s} \). Typical lab experiments involve flux changes in the µWb to mWb range, while power transformers work with much larger fluxes.
What is the difference between average and instantaneous EMF?
The calculator returns the average EMF over your time interval: \( N\Delta\Phi/\Delta t \). If the flux changes at a constant rate, this equals the instantaneous value \( N|d\Phi/dt| \) at every moment. If the change is non-uniform — for example a sinusoidal flux in an AC generator — the instantaneous EMF peaks higher than the average, so don't use the average value as a peak voltage in circuit design.
How do I find the direction of the induced current?
Apply Lenz's law: the induced current flows so that its own magnetic field opposes the change in flux. If the flux through the coil is increasing, the current creates a field opposing the external one; if the flux is decreasing (magnet pulled away), the current reinforces it. Viewed from the side the magnet approaches, the current appears counterclockwise; from the side it leaves, clockwise.
What are the most common mistakes in Faraday's law problems?
The top errors are mixing unit prefixes (entering 5 mWb as 5 Wb gives a result 1,000× too large), forgetting to take the absolute difference \( \Delta\Phi = |\Phi_2 - \Phi_1| \), using the number of turns as 1 when the coil has many turns, and plugging in milliseconds as seconds. Also remember that a single turn cutting through 1 Wb in 1 s yields only 1 V — most practical voltages come from hundreds of turns.